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G = D4.5D8  order 128 = 27

5th non-split extension by D4 of D8 acting via D8/C8=C2

p-group, metabelian, nilpotent (class 4), monomial

Aliases: D4.5D8, Q8.5D8, C16.24D4, M4(2).36D4, M5(2).28C22, D4○C16⋊2C2, C4.43(C2×D8), (C2×SD32)⋊2C2, C4○D4.28D4, C8.6(C4○D4), C8.107(C2×D4), D4.4D4.C2, C8.4Q8⋊7C2, M5(2)⋊C2⋊8C2, D4.5D4⋊3C2, C8.17D4⋊8C2, C8○D4.9C22, C4.98(C4⋊D4), C2.26(C8⋊7D4), (C2×C8).240C23, (C2×C16).28C22, (C2×D8).50C22, C22.8(C4○D8), C8.C4.7C22, (C2×Q16).49C22, (C2×C4).45(C2×D4), SmallGroup(128,955)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C8 — D4.5D8
Chief series
C1 — C2 — C4 — C2×C4 — C2×C8 — C8○D4 — D4○C16 — D4.5D8
Lower central
C1 — C2 — C4 — C2×C8 — D4.5D8
Upper central
C1 — C2 — C2×C4 — C8○D4 — D4.5D8
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — D4.5D8

Generators and relations for D4.5D8
 G = < a,b,c,d | a4=b2=1, c8=d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c7 >

Subgroups: 180 in 72 conjugacy classes, 30 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C16, C16, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C4○D4, C4.D4, C4.10D4, C8.C4, C2×C16, C2×C16, M5(2), M5(2), SD32, C8○D4, C2×D8, C2×Q16, C8⋊C22, C8.C22, M5(2)⋊C2, C8.17D4, C8.4Q8, D4.4D4, D4.5D4, D4○C16, C2×SD32, D4.5D8
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C4○D4, C4⋊D4, C2×D8, C4○D8, C8⋊7D4, D4.5D8

Character table of D4.5D8

 class 12A2B2C2D4A4B4C4D8A8B8C8D8E8F8G16A16B16C16D16E16F16G16H16I16J
 size 112416224162244416162222444444
ρ111111111111111111111111111    trivial
ρ2111-1-111-1-1111-1-11111111-1-1-1-11    linear of order 2
ρ3111-1-111-11111-1-1-11-1-1-1-1-11111-1    linear of order 2
ρ411111111-111111-11-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ51111-11111111111-1-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ6111-1111-1-1111-1-11-1-1-1-1-1-11111-1    linear of order 2
ρ7111-1111-11111-1-1-1-111111-1-1-1-11    linear of order 2
ρ81111-1111-111111-1-11111111111    linear of order 2
ρ922-2002-20022-20000-2-2-2-2200002    orthogonal lifted from D4
ρ10222202220-2-2-2-2-2000000000000    orthogonal lifted from D4
ρ1122-2002-20022-200002222-20000-2    orthogonal lifted from D4
ρ12222-2022-20-2-2-222000000000000    orthogonal lifted from D4
ρ1322-220-22-200000000-√2√2-√2√2-√2-√2√2-√2√2√2    orthogonal lifted from D8
ρ1422-2-20-22200000000√2-√2√2-√2√2-√2√2-√2√2-√2    orthogonal lifted from D8
ρ1522-220-22-200000000√2-√2√2-√2√2√2-√2√2-√2-√2    orthogonal lifted from D8
ρ1622-2-20-22200000000-√2√2-√2√2-√2√2-√2√2-√2√2    orthogonal lifted from D8
ρ1722200-2-2000002i-2i00√2-√2√2-√2-√2-√-2√-2√-2-√-2√2    complex lifted from C4○D8
ρ1822-2002-200-2-220000000002i2i-2i-2i0    complex lifted from C4○D4
ρ1922200-2-200000-2i2i00√2-√2√2-√2-√2√-2-√-2-√-2√-2√2    complex lifted from C4○D8
ρ2022200-2-2000002i-2i00-√2√2-√2√2√2√-2-√-2-√-2√-2-√2    complex lifted from C4○D8
ρ2122-2002-200-2-22000000000-2i-2i2i2i0    complex lifted from C4○D4
ρ2222200-2-200000-2i2i00-√2√2-√2√2√2-√-2√-2√-2-√-2-√2    complex lifted from C4○D8
ρ234-400000002√2-2√2000002ζ1613+2ζ16112ζ1615+2ζ1692ζ165+2ζ1632ζ167+2ζ16000000    complex faithful
ρ244-400000002√2-2√2000002ζ165+2ζ1632ζ167+2ζ162ζ1613+2ζ16112ζ1615+2ζ169000000    complex faithful
ρ254-40000000-2√22√2000002ζ1615+2ζ1692ζ165+2ζ1632ζ167+2ζ162ζ1613+2ζ1611000000    complex faithful
ρ264-40000000-2√22√2000002ζ167+2ζ162ζ1613+2ζ16112ζ1615+2ζ1692ζ165+2ζ163000000    complex faithful

Smallest permutation representation of D4.5D8
►On 32 points
Generators in S32
(1 13 9 5)(2 14 10 6)(3 15 11 7)(4 16 12 8)(17 21 25 29)(18 22 26 30)(19 23 27 31)(20 24 28 32)

(1 28)(2 29)(3 30)(4 31)(5 32)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)(12 23)(13 24)(14 25)(15 26)(16 27)

(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32)

(1 8 9 16)(2 15 10 7)(3 6 11 14)(4 13 12 5)(17 18 25 26)(19 32 27 24)(20 23 28 31)(21 30 29 22)
 
G:=sub<Sym(32)| (1,13,9,5)(2,14,10,6)(3,15,11,7)(4,16,12,8)(17,21,25,29)(18,22,26,30)(19,23,27,31)(20,24,28,32), (1,28)(2,29)(3,30)(4,31)(5,32)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(12,23)(13,24)(14,25)(15,26)(16,27), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32), (1,8,9,16)(2,15,10,7)(3,6,11,14)(4,13,12,5)(17,18,25,26)(19,32,27,24)(20,23,28,31)(21,30,29,22)>;
 
G:=Group( (1,13,9,5)(2,14,10,6)(3,15,11,7)(4,16,12,8)(17,21,25,29)(18,22,26,30)(19,23,27,31)(20,24,28,32), (1,28)(2,29)(3,30)(4,31)(5,32)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(12,23)(13,24)(14,25)(15,26)(16,27), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32), (1,8,9,16)(2,15,10,7)(3,6,11,14)(4,13,12,5)(17,18,25,26)(19,32,27,24)(20,23,28,31)(21,30,29,22) );
 
G=PermutationGroup([[(1,13,9,5),(2,14,10,6),(3,15,11,7),(4,16,12,8),(17,21,25,29),(18,22,26,30),(19,23,27,31),(20,24,28,32)], [(1,28),(2,29),(3,30),(4,31),(5,32),(6,17),(7,18),(8,19),(9,20),(10,21),(11,22),(12,23),(13,24),(14,25),(15,26),(16,27)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)], [(1,8,9,16),(2,15,10,7),(3,6,11,14),(4,13,12,5),(17,18,25,26),(19,32,27,24),(20,23,28,31),(21,30,29,22)]])
 

Matrix representation of D4.5D8 ►in GL4(𝔽7) generated by

0651
3056
3361
1631
,
4320
6046
1145
0006
,
2045
5235
5214
4453
,
3633
6525
6102
2406
G:=sub<GL(4,GF(7))| [0,3,3,1,6,0,3,6,5,5,6,3,1,6,1,1],[4,6,1,0,3,0,1,0,2,4,4,0,0,6,5,6],[2,5,5,4,0,2,2,4,4,3,1,5,5,5,4,3],[3,6,6,2,6,5,1,4,3,2,0,0,3,5,2,6] >;
 

D4.5D8 in GAP, Magma, Sage, TeX 

D_4._5D_8
 
% in TeX
 
G:=Group("D4.5D8");
 
// GroupNames label
 
G:=SmallGroup(128,955);
 
// by ID
 
G=gap.SmallGroup(128,955);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,736,422,1123,360,2804,718,172,4037,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^2=1,c^8=d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^7>;
 
// generators/relations
 

Export

Character table of D4.5D8 in TeX

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